As noted, there are primes missing, so they do not yield all of the prime powers of 2.
Even the sequence of all integers employs "in part" the numbers included in this mapping. Your wording describes a sequence that includes numbers in addition to those generated by the mapping. Did you mean it to be interpreted differently?
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davilla
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davilla said:
Name the sequence generated employing, in part, a mapping of h -> to the first sequence in the problem.
What I mean is, find the significance of the first sequence. To that end, I have provided a formula that will prove useful, without giving too much away.